diff --git a/xarray/core/indexes.py b/xarray/core/indexes.py
index 844751f24bb..b66fbdf6504 100644
--- a/xarray/core/indexes.py
+++ b/xarray/core/indexes.py
@@ -231,9 +231,14 @@ def query(self, labels, method=None, tolerance=None):
                         )
                     indexer = self.index.get_loc(label_value)
                 else:
-                    indexer = self.index.get_loc(
-                        label_value, method=method, tolerance=tolerance
-                    )
+                    if method is not None:
+                        indexer = get_indexer_nd(self.index, label, method, tolerance)
+                        if np.any(indexer < 0):
+                            raise KeyError(
+                                f"not all values found in index {coord_name!r}"
+                            )
+                    else:
+                        indexer = self.index.get_loc(label_value)
             elif label.dtype.kind == "b":
                 indexer = label
             else:
diff --git a/xarray/core/missing.py b/xarray/core/missing.py
index 2525272f719..f3bb5351db5 100644
--- a/xarray/core/missing.py
+++ b/xarray/core/missing.py
@@ -564,9 +564,8 @@ def _localize(var, indexes_coords):
         minval = np.nanmin(new_x.values)
         maxval = np.nanmax(new_x.values)
         index = x.to_index()
-        imin = index.get_loc(minval, method="nearest")
-        imax = index.get_loc(maxval, method="nearest")
-
+        imin = index.get_indexer([minval], method="nearest").item()
+        imax = index.get_indexer([maxval], method="nearest").item()
         indexes[dim] = slice(max(imin - 2, 0), imax + 2)
         indexes_coords[dim] = (x[indexes[dim]], new_x)
     return var.isel(**indexes), indexes_coords